Q. Two rigid boxes containing different ideal gases are placed on a table. Box $A$ contains one mole of nitrogen at temperature $T_0$, while box $B$ contains one mole of helium at temperature $(7/3)\, T_0$. The boxes are then put into thermal contact with each other, and heat flows between them until the gases reach a common final temperature (Ignore the heat capacity of boxes). Then, the final temperature of the gases, $T_f$, in terms of/$T_0$ is :
AIEEEAIEEE 2006Thermal Properties of Matter
Solution:
Here, change in internal energy of the system is •icro. i.e., increase in internal energy of one is equal to decrease in internal energy of other.
$\Delta U_{A}=1\times\frac{5R}{2}\left(T_{f} -T_{0}\right)$
$\Delta U_{B}=1\times\frac{3R}{2}\left(T_{f} -\frac{7}{3}T_{0}\right)$
Now $\Delta U_{A}+\Delta U_{B}=0$
$\frac{5R}{2}\left(T_{f} -T_{0}\right)+\frac{3R}{2}\left(T_{f} -\frac{7T_{0}}{3}\right)=0$
$5T_{f} -5T_{0}+3T_{f} -7T_{0}=0$
$\Rightarrow 8T_{f} =12T_{0} \Rightarrow T_{f} =\frac{12}{8}T_{0}=\frac{3}{2}T_{0}$
